Local Complexity of Finite Semigroups

Abstract

This chapter discusses local complexity of finite semigroups. The complexity function c: TS → N, which assigns to every finite transformation semigroup X a nonnegative integer Xc, is the largest function d: TS → N. The meaning of X <Y for machines is that Y is capable of computing whatever X can compute. A machine cannot be less complex than a machine it dominates. Under series composition, the complexity cannot rise faster than the sum of the parts, whereas parallel composition does not increase complexity at all. The chapter also presents conditions under which a function can be called a local complexity function.